Hopper Designing

8/18/2019 Hopper Designing

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F 4.1


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Flow of Particulate Solids in Bunkers and Flow Problems

Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect

(Expanded Flow)

Numbers show the sequence of discharge of bulk layers

height

levels of

free bulk

surface velocity

profiles5

7

6

4

3

2

8

1

Θ

1

7

6

8

5

4

3

2

1Θ

ϕb

ϕb

3 3

4 42 2

5 5

6

7

6

71 1

8 8

Θ1

Θ2

plug

flow

angle of

repose

dead

zones

Θ

Channelling, Piping, Ratholing Bridging, Arching

dead

zones

ΘΘ

F 4.2


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F 4.3


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Dynamics of Force Balance at Cohesive Powder Bridge

B

Θ

dFT

h

Θ

W

1́dFV

b

dFf

VF

dFV

dFGdhB

1́

slot length l

Dead weight of powder bridge

Wall force

Force of inertia

Drag force of penetrating fluid

F = 0 = - dFG + dFT + dFV + dFf

dFG = b g b dhB l. . . .

dFV = 1' sin dhB cos 2l. . . .

dFT = dFG .a

g

dFf = Eu b l dhB .3 f u

2 (1 - )

4 d 2

. . .

. ....

F 4.4


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1. Mass Flow

- Avoid Channelling:

Hopper angle = f(wall friction angle W, effektive angle of internalfriction e)

see diagrams F 4.6 and F 4.7

- Avoid Bridging:

1.1 Free Flowing Bulk Solid (avoid machanical blocking of coarse lumps or rocks):

σc,crit critical uniaxial compressive strength

ρb,crit bulk density at σ1,crit

g gravitational acceleration

article size

k = 0.6 ... 1.4 shape dependent parameter

bmin

1.2 Cohesive Powder (avoid cohesive bridges):

- Effective wall stress at arch: ´ = 1 /ff (2)

- Flow factor (diagram F 4.11): ff = f( e, W, ) (3)

(4)

(1a)

(1b)

Apparatus Design of Silo Hopper to Avoid Bridging

F 4.5

slot width (1c)

bmin

= + W

b · g · b

1́ 1́


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0 10 20 30 40 50 60

45

40

35

30

25

20

15

10

5

0

hopper angle versus vertical in deg

a n g l e o f w a l l f r

i c t i o n

w

i n d e g

Mass Flow

Core Flow

effective angle of

internal friction

e = 70° 60° 50° 40° 30°

12

180° - arccos1 - sin e2 sin e

- W - arc sin sin W sin e

Bounds between Mass and Core Flowaxisymmetric Flow

(conical hopper)

select

F 4.6


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50

45

40

35

30

25

20

15

10

5

0

a

n g l e o f w a l l f r i c t i o n

w

i n d e g

55

0 10 20 30 40 50 60

hopper angle versus vertical in deg

Core Flow

effective angle of internal friction

e = 70° 60° 50° 40° 30°

Mass Flow

60,5° +

arc tan50° - e7,73°

15,07°1-

42,3° + 0,131° · exp(0,06 · e)

W

with W 3° ande 60°

Bounds between Mass and Core Flow

Plane Flow(wedge-shaped hopper)

F 4.7


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max

l m i n > 3 · b

m i n

b m i n

b m i n

D

l m i n > 3 · b

m i n

bmin

max maxwall

b m i n

- Conical Hopper (axisymmetric stress field)

Cone Pyramid

shape factor m = 1 [ 3a ]

- Wedge-shaped Hopper (plane stress field)

vertical front walls

shape factor m = 0

F 4.8


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inclined front walls

1 , 5 b m i n

b m i n

l m i n > 6 · b

m i n

3 b m i n

B

L

1max

2max

1 , 5 b m i n

F 4.9


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u n c o n f i n e d y i e l d s t r e n g t h

c

c,0

major principal stress during

consolidation (steady-state flow) 1

0

c = a1 · 1 + c,0

e f f e

c t i v e w a l l s t r e s s

'

' = 1 / ff 1

bmin 1'1'

c,crit

uniaxial compressive strength c

' c flow

' c stable arch

' c,crit

Arching/Flow Criterion of a Cohesive Powder in a Convergent Hopper

F 4.10


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20 30 40 50 60 70

1,5

f l o w f a c t o r

f f

effective angle of internal friction e in deg

2

1

conical hopper

wedge-shaped hopper

Ascertainment of Approximated Flow Factor

(angle of wall friction W = 10° - 30°)

F 4.11


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F 4.12


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b u l k d e n s

i t y

b

b,0

*

90°

1

1

u n c o n f i n e d y

i e l d s t r e n g t h

c

1 = c,st

ff = 1

c,0

c,st

major principal stress during

consolidation (steady-state flow) 1

0

c = a1 · 1 + c,0

a n g l e s o f i n t e r n a l

f r i c t i o n

e ,

s t ,

i

e f f e c t i v e w a l l

s t r e s s

'

b,crit

b,st

' = 1 / ff 1

bmin

1'

bmin,st

1'

stationary angle of internal friction st = const.

angle of internal friction i ≈ const.

effective angle of internal friction e

uniaxial compressive strength c

bulk density b

c,crit

Consolidation Functions of a Cohesive Powder for Hopper Design for Reliable Flow

F 4.13

0


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F 4.15


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F 4.16


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(9a)

pv

CF

bC,min

G (angle of internal friction i or it) - function, see F 4.22

Vertical pressure at filling, F 4.20:

1 pv = f ( e, W, b, shaft cross section,

silo height) (8a)

c,crit see F 4.19

≈

a) Maximum approach at filling and consolidation:

F 4.172. Core Flow

Avoid channelling (stable funnel)

Hopper angle

2.1 Free Flowing Bulk Solid see 1.1

2.2 Cohesive Powder


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2. Core Flow - Supplement

Avoid channelling (stable funnel)

Hopper angle: W

2.1 Free Flowing Bulk Solid see 1.1

2.2 Cohesive Powder

bC,min

AA

ChannelA - A: Ring stress 1'' at surface of channel wall

1'' 1''

bC,min

G (Angle of internal friction i or it) - function, see F 4.22

b) Filling, consolidation and

anisotropy1)

:Horizontal pressure at filling, F 4.20:

1'' ph = f ( e, W, b, shaft cross section silo height)

≈(8b)

(9b)

c) Flow and radial stress field,F 4.10, Ring stress:

(8c)1'' = 1ff d

Flow factor of channelling:

(8d)

Two additional options:

F 4.18


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ct

a n g l e o f w a l l f r i c t i o n

w

s t a

t i o n a r y a n g l e o f

i n

t e r n a l f r i c t i o n

s t

b u l k d e n s i t y

ρ b

mass flow hopper

core flow hopper

major principal stress 1

a n g l e o f i n t e r n a l f r i c t i o n

i a n d

i t

b

e st

it

i

w

u n i a x i a l c o m p r e s s i v e s t r e n g t h

c

e f f e c t i v e w a l l s t r e s s

1`

c

1́

1

1

1

1

Consolidation Functions of Cohesive Powders for Hopper Design

c,crit(core flow)

c,crit

ct,crit(mass flow)

ct,crit(core flow)

e f f e c t i v e a n g l e o f

i n

t e r n a l f r i c t i o n

s t

F 4.19


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Calculation of Silo Pressures according to Slice-Element Method

Force Balance F = 0

Shaft (Filling F):

H

H T r

pv

pv

pnpn pW

pWdA

y

y

pWpW

d y

d y

ph ph

H*

b · g · dy

b · g · dy

pv + dpv

pv + dpv

Hopper:

F 4.20


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f u n c t i o n

G

i )

0 10 20 30 40 50 60 70 80

angle of internal frictioni

in deg

10

9

8

7

6

5

4

3

2

1

0

Function G( i) to Design a Hopper for Core Flow

F 4.22


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Estimation of Minimum Shaft Diameter

Process Parameters and Geometrical Apparatus Parameters

pressures p

h e i g t h H

pW

pv

ph

shaft diameter Dmin

h e i g h t H

a) Calculation of vertical pressure

Filling /Storage

b) Consolidation function

c

1

c,0

c) Shaft design equation

D H

b

or

F 4.23

a =1 - sin2 w1 + sin2 w+

- (1 - sin2 w).(sin2 e - sin

2w)

(1 - sin2 w).(sin2 e - sin

2w)

(1a) (1b)

(1c)

(2)

(3)

(4)

(5)


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detail "Z"

maximum roof loads:filter load: 6 kN

snow load: 1 kN /m2

gangway:walking monoload: 1,5 kN

evenly distributed: 0,75 kN/m2

h 2

h 3

18d3Fl 100 x15

name and rated width of support

ratedvolume

Vm3

i n p u t

o u t p u t N D 6

T G L 0 - 2 5 0 1

b y - p a s s

f i l t e r l i n k

F T F N

r e s e r v e

w o r k o p e n i n g

l e v e l i n d i c a t i o n

s a f e t y d e v i c e

l i f t i n g a r m

~ T G L 3 1 - 4 6 1

c a r r i e r e y e

~ T G L 3 1 - 3 4 3

20

40

80

100

160320

100 200 200 600 600 150/50 200 B 160 A 300

250

300

893 x666

B 90

B 110

B 220B 325

A 250

-

p1 p2 p3 p4 p5 r1 r2 s1 s2 t2t1

3000

5000

30755080

24

36

d1) d3

n u m b e r o f

b o l t s

w o r k o p e n i n g

20

40

80

100160

320

ratedvolume

Vm3

d1) R1 R2 1 2

[ °] [ °]

h1 h2 h4 h5 h7h3 h9

=30°

mass2)

kg

3000

3000

3000

5000

5000

3000

775 1050 35 40 325 1820 750

1750 1550 25 30 420 3200 - 900300

200

150

350

1130

1550

2990

34803875

8180

300

800

2800

4290

3000

6000

11000

140007000

16000

59208920

13920

169201187020870

1) d = vessel outer diameter2) total mass for Al Mg 3 ( sS = 2,7 t / m

3)

=30°

Standard Silo

earthing

F 4.24


25/34

Comparison of Models to Calculate theHopper Discharge Mass Flow Rate

valid for: consider:

cohesion-less

hoppershape

flowcondi-tions

a i r d r a g

p r e s s u r e

d e p e n d e n c y o f

c o h e s i v e

F 4.25


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F 4.26


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Stationary Discharge Flow Rate versus Particle Size for Sand

conical mass flow hopper

1,5

1,4

1,3

1,2

1,1

1,0

0,9

0,8

0,7

0,60,5

0,4

0,3

0,2

0,1

0

d i s c h a r g e f l o w r a t e v s

i n m / s

b = 0,156 m

= 10°

b = 0,036 m

= 10°b = 0,036 m

= 15°

b = 0,0167 m= 10°

b = 0,0103 m

= 10°

calculated (Tomas)measured (Carleton)

5 10 -2 2 5 10-1 2 5 10 0 2 5 101 2 5 102

particle size d in mm

k =3, = 1, ff > 10b c

F 4.27


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F 4.29


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Equipment for Filling of Silos

- to avoid segregation

F 4.30


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Methods to Control the Level of Silos

1. Pressure gauges 2. Mechanical plumb

3. Revolving blade devices

4. Membrane pressure switch

5. Conductivity measurement

6. Capacity measurement

7. Radiometric measurement 8. Ultra-sonic measurement

F 4.31


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bladetype

material installationlength in m

type

N

St

N C - 0,4 - 0,14 - NSt C - 0,4 - 0,14 - St

N C - 0,4 - 0,36 - NSt C - 0,4 - 0,36 - St

N C - 0,4 - 0,11 - NSt C - 0,4 - 0,11 - St

0,250,51,0

0,250,51,0

0,4

bendedprotection

pipe

C - 0,25 - 0,14 - NC - 0,5 - 0,14 - NC - 1,0 - 0,14 - N

C - 0,25 - 0,14 - StC - 0,5 - 0,14 - StC - 1,0 - 0,14 - St

1 4 5

0,14 C

145

0,14

360

0,36

110

∅ 1

0

0,11

Revolving Blade Level Indicator LS 40

LS 40/A - 0,1 toLS 40/A - 3,0

normal edition

LS 40/B - 0,25 toLS 40/B - 6,0

with protection pipe from

carbon (St) or

stainless steel (N)

LS 40/C - 0,25 toLS 40/C - 1,0

LS 40/C - 0,4 - 0,14

installation atinclined wall

r a t e d l

e n g t h

r a t e d

l e n g t h

F 4.32


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Hopper Locks

horizontal gate vertical gate horizontal rotaryslide-valve

double rotaryslide-valve

ball valve rotary disk valve

discharge chute withclaw lever lock

lock with swivel chute

F 4.33


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Size in mm h1 h2 l1 l2 l3 Mass P

in kg in kW 250 120 136 1245 905 180 200

315 1450 1045 217 230

400 140 1735 1235 265 260500 119 2050 1445 317 325

630 2405 1685 380 410

800 2915 2025 465 535

1000 180 101 3530 2435 570 785

Hopper gates with drive

118

1111600.75

1.1

b1 see table above

0.55

Size in mm b1 d1 h1 h2 l1 l2 l3 Mass in kg

250 250 120 86 1097 982 180 70

315 315 1230 1115 218 92

400 410 315 140 100 1420 1305 265 123

500 515 1630 1515 318 147

630 630 1925 1810 380 221 800 800 400 160 114 2652 2362 465 393

1000 1000 180 132 3100 2810 570 570

Hopper gates

F 4.34

Hopper Designing

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